The Hellinger-Kantorovich (HK) distance is an unbalanced extension of the Wasserstein-2 distance. It was shown recently that the HK barycenter exhibits a much more complex behaviour than the Wasserstein barycenter. Motivated by this observation we study the HK barycenter in more detail for the case where the input measures are an uncountable collection of Dirac measures, in particular the dependency on the length scale parameter of HK, the question whether the HK barycenter is discrete or continuous and the relation between the expected and the empirical barycenter. The analytical results are complemented with numerical experiments that demonstrate that the HK barycenter can provide a coarse-to-fine representation of an input pointcloud or measure.
翻译:Hellinger-Kantorovich(HK)距离是Wasserstein-2距离的不平衡延伸,最近显示,香港的百居中心展示的行为比瓦森斯坦的百居中心复杂得多。我们受此观察的启发,更详细地研究了香港的百居中心,研究的是输入措施是无法量化的Dirac衡量标准集合,特别是对香港长度参数的依赖,香港的百居中心是离散还是连续的,以及预期的和实证的百居中心之间的关系。分析结果还辅以数字实验,表明香港的百居中心可以提供粗略的输入点或测量标准。